Buckling of Axisymmetric, Homogeneous Cylindrical Shells With Random Imperfections: Monte Carlo Method

2005 ◽  
Author(s):  
Dennis Williams
Keyword(s):  
Monte Carlo ◽  
Cylindrical Shell ◽  
Pressure Vessel ◽  
Cylindrical Shells ◽  
Large Diameter ◽  
Buckling Load ◽  
Problem Solution ◽  
Critical Buckling Load ◽  
Asme Code ◽  
Section Viii

This paper presents the second of a series of solutions to the buckling of imperfect cylindrical shells subjected to an axial compressive load. In particular, the current problem reviewed is the case of a homogeneous cylindrical shell with random axisymmetric imperfections. The problem solution for the determination of the critical buckling load utilizes a statistical approach to define the random imperfections as opposed to the deterministic methods most often employed in the pressure vessel industry. The imperfections are treated as a random function of the axial (i.e., longitudinal) position on the shell. The Monte Carlo technique is utilized to create a large sample of random shell geometries from which to eventually calculate a critical buckling load for each randomly generated shell geometry. Having matched or predefined the statistical parameters (including the co-variance) of interest as determined from actual manufacturing statistics to the Monte Carlo simulation of shell geometries, the reliability of the critical buckling load is then calculated for the set of cylindrical shells with the random axisymmetric imperfections. The ASME Boiler and Pressure Vessel Code Section VIII fabrication tolerances as supplemented by ASME Code Case 2286-1 are reviewed and addressed in light of the findings of the current study and resulting solutions with respect to the critical buckling loads. The method and results described herein are in stark contrast to the “knockdown factor” approach currently utilized in ASME Code Case 2286-1. Recommendations for further study of the imperfect cylindrical shell are also outlined in an effort to improve on the current design rules regarding column buckling of large diameter shells designed in accordance with ASME Section VIII, Divisions 1 and 2 and ASME STS-1 in combination with the suggestions contained within Code Case 2286-1.

Fourier Series Analysis of Cylindrical Pressure Vessel Subjected to External Pressure

2009 ◽  
Author(s):  
Gurinder Singh Brar ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Fourier Series ◽  
Pressure Vessel ◽  
Large Diameter ◽  
External Pressure ◽  
Pressure Vessels ◽  
Buckling Load ◽  
Critical Buckling Load ◽  
Cylindrical Pressure Vessel ◽  
Section Viii ◽  
Division 2

This paper presents the comparison of a reliability technique that employs a Fourier series representation of random asymmetric imperfections in a cylindrical pressure vessel subjected to external pressure. Comparison with evaluations prescribed by the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 Rules for the same shell geometries are also conducted. The ultimate goal of the reliability type technique is to predict the critical buckling load associated with the chosen cylindrical pressure vessel. Initial geometric imperfections are shown to have a significant effect on the load carrying capacity of the example cylindrical pressure vessel. Fourier decomposition is employed to interpret imperfections as structural features that can be easily related to various other types of defined imperfections. The initial functional description of the imperfections consists of an axisymmetric portion and a deviant portion, which are availed in the form of a double Fourier series. Fifty simulated shells generated by the Monte Carlo technique are employed in the final prediction of the critical buckling load. The representation of initial geometrical imperfections in the cylindrical pressure vessel requires the determination of appropriate Fourier coefficients. Multi-mode analyses are expanded to evaluate a large number of potential buckling modes for both predefined geometries and associated asymmetric imperfections as a function of position within a given cylindrical shell. The probability of the ultimate buckling stress that may exceed a predefined threshold stress is also calculated. The method and results described herein are in stark contrast to the “knockdown factor” approach as applied to compressive stress evaluations currently utilized in industry. Recommendations for further study of imperfect cylindrical pressure vessels are also outlined in an effort to improve on the current design rules regarding column buckling of large diameter pressure vessels designed in accordance with ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 and ASME STS-1.

scholarly journals Buckling of Imperfect, Axisymmetric, Homogeneous Shells of Variable Thickness: Perturbation Solution

2005 ◽  
Author(s):  
Dennis Williams
Keyword(s):  
Cylindrical Shell ◽  
Variable Thickness ◽  
Large Diameter ◽  
Compressive Load ◽  
Thin Wall ◽  
Shells Of Revolution ◽  
Equilibrium Equations ◽  
Asme Code ◽  
Current Design ◽  
Section Viii

This paper presents the first of a series of solutions to the buckling of imperfect cylindrical shells subjected to an axial compressive load. In particular, the initial problem reviewed is the case of a homogeneous cylindrical shell of variable thickness that is of an axisymmetric nature. The equilibrium equations as first introduced by Donnell over seventy years ago are thoroughly presented as a basis for embarking upon a solution that makes use of perturbation methods. The ultimate objective of these calculations is to achieve a quantitative assessment of the critical buckling load considering the small axisymmetric deviations from the nominal shell wall thickness. Clearly in practice, large diameter, thin wall shells of revolution that form stacks (as found in flue gas desulphurization absorber assemblies) are never fabricated with constant diameters and thicknesses over the entire length of the assembly. As such, ASME Boiler and Pressure Vessel Code Section VIII fabrication tolerances as supplemented by ASME Code Case 2286-1 are reviewed and addressed in light of the findings of the current study and resulting solutions with respect to the critical buckling loads. The method and results described herein are in stark contrast to the “knockdown factor” approach currently utilized in ASME Code Case 2286-1. Recommendations for further study of the imperfect cylindrical shell are also outlined in an effort to improve on the current design rules regarding column buckling of large diameter shells designed in accordance with ASME Section VIII, Divisions 1 and 2; and ASME STS-1 in combination with the suggestions contained within Code Case 2286-1.

Buckling of Axisymmetric Cylindrical Shells of Variable Thickness: Finite Difference Solution

2007 ◽  
Author(s):  
Gurinder Singh Brar ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Cylindrical Shell ◽  
Finite Difference ◽  
Variable Thickness ◽  
Cylindrical Shells ◽  
Finite Difference Methods ◽  
Large Diameter ◽  
Buckling Load ◽  
Thin Wall ◽  
Shells Of Revolution ◽  
Equilibrium Equations

This paper presents the third of a series of solutions to the buckling of imperfect cylindrical shells subjected to an axial compressive load. In particular, the initial problem reviewed is the case of a homogeneous cylindrical shell of variable thickness that is of an axisymmetric nature. The equilibrium equations as first introduced by Donnell over seventy years ago are discussed and reviewed in establishing a basis for embarking upon a solution that utilizes finite difference methods to solve the resulting equilibrium and compatibility equations. The ultimate objective of these calculations is to achieve a quantitative assessment of the critical buckling load considering the small axisymmetric deviations from the nominal cylindrical shell wall thickness. Clearly in practice, large diameter, thin wall shells of revolution that form stacks are never fabricated with constant diameters and thicknesses over the entire length of the assembly. The method and results described herein are in stark contrast to the “knockdown factor” approach currently utilized in ASME Code Case 2286-1. The results obtained by finite difference method agree well with those published by Elishakoff and Williams for the prediction of buckling load.

Calculation of Working Pressure for Cylindrical Vessel Under External Pressure

2010 ◽  
Author(s):  
Gurinder Singh Brar ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Monte Carlo ◽  
Cylindrical Shell ◽  
Pressure Vessel ◽  
External Pressure ◽  
Working Pressure ◽  
Geometric Imperfections ◽  
The Monte Carlo Method ◽  
Cylindrical Pressure Vessel ◽  
Section Viii ◽  
Initial Geometric Imperfections

Initial geometric imperfections have a significant effect on the load carrying capacity of asymmetrical cylindrical pressure vessels. This research paper presents a comparison of a reliability technique that employs a Fourier series representation of random asymmetric imperfections in a defined cylindrical pressure vessel subjected to external pressure. Evaluations as prescribed by the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 rules are also presented and discussed in light of the proposed reliability technique presented herein. The ultimate goal of the reliability type technique is to statistically predict the buckling load associated with the cylindrical pressure vessel within a defined confidence interval. The example cylindrical shell considered in this study is a fractionating tower for which calculations have been performed in accordance with the ASME B&PV Code. The maximum allowable external working pressure of this tower for the shell thickness of 0.3125 in. is calculated to be 15.1 psi when utilizing the prescribed ASME B&PV Code, Section VIII, Division 1 methods contained within example L-3.1. The Monte Carlo method as developed by the current authors and published in the literature is then used to calculate the maximum allowable external working pressure. Fifty simulated shells of geometry similar to the example tower are generated by the Monte Carlo method to calculate the nondeterministic buckling load. The representation of initial geometric imperfections in the cylindrical pressure vessel requires the determination of appropriate Fourier coefficients. The initial functional description of the imperfections consists of an axisymmetric portion and a deviant portion that appears in the form of a double Fourier series. Multi-mode analyses are expanded to evaluate a large number of potential buckling modes for both predefined geometries and the associated asymmetric imperfections as a function of position within a given cylindrical shell. The method and results described herein are in stark contrast to the dated “knockdown factor” approach currently utilized in ASME B&PV Code.

Buckling of Imperfect, Axisymmetric, Homogeneous Shells of Variable Thickness: Perturbation Solution

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Dennis K. Williams ◽  
James R. Williams ◽  
Yogeshwar Hari
Keyword(s):  
Cylindrical Shell ◽  
Variable Thickness ◽  
Large Diameter ◽  
Compressive Load ◽  
Thin Wall ◽  
Shells Of Revolution ◽  
Equilibrium Equations ◽  
Asme Code ◽  
Section Viii ◽  
Ultimate Objective

This paper presents the first of a series of solutions to the buckling of imperfect cylindrical shells subjected to an axial compressive load. The initial problem is the case of a homogeneous cylindrical shell of variable thickness that is of an axisymmetric nature. The equilibrium equations as first introduced by Donnell over 70 years ago are thoroughly presented as basis for a solution employing perturbation methods. The ultimate objective of these calculations is to achieve a quantitative assessment of the critical buckling load considering the small axisymmetric deviations from the nominal shell wall thickness. Clearly in practice, large diameter thin wall shells of revolution that form stacks (as found in flue gas desulphurization absorber assemblies) are never fabricated with constant diameters and thicknesses over the entire length of the assembly. As such, ASME Boiler and Pressure Vessel Code Section VIII shell thickness tolerances as supplemented by ASME Code Case 2286-1 are reviewed and addressed in comparison to the resulting solutions with respect to the critical buckling loads. The method and results described herein are in stark contrast to the “knockdown factor” approach currently utilized in ASME Code Case 2286-1. Recommendations for further study of the imperfect cylindrical shell are also outlined in an effort to improve on the current rules regarding column buckling of large diameter shells designed in accordance with ASME Section VIII, Divisions 1 and 2 and ASME STS-1 in combination with the suggestions contained within Code Case 2286-1.

Buckling of 2D-FG Cylindrical Shells under Combined External Pressure and Axial Compression

2013 ◽  
Vol 5 (03) ◽  
pp. 391-406 ◽  
Author(s):  
R. Mohammadzadeh ◽  
M. M. Najafizadeh ◽  
M. Nejati
Keyword(s):  
Cylindrical Shell ◽  
Axial Compression ◽  
Cylindrical Shells ◽  
Axial Direction ◽  
External Pressure ◽  
Functionally Graded ◽  
Buckling Load ◽  
Critical Buckling Load ◽  
Classical Shell Theory ◽  
The Stability

AbstractThis paper presents the stability of two-dimensional functionally graded (2D-FG) cylindrical shells subjected to combined external pressure and axial compression loads, based on classical shell theory. The material properties of functionally graded cylindrical shell are graded in two directional (radial and axial) and determined by the rule of mixture. The Euler’s equation is employed to derive the stability equations, which are solved by GDQ method to obtain the critical mechanical buckling loads of the 2D-FG cylindrical shells. The effects of shell geometry, the mechanical properties distribution in radial and axial direction on the critical buckling load are studied and compared with a cylindrical shell made of 1D-FGM. The numerical results reveal that the 2D-FGM has a significant effect on the critical buckling load.

Pasternak effect on the buckling of embedded single-walled carbon nanotubes using non-local cylindrical shell theory

2011 ◽  
Vol 225 (12) ◽  
pp. 3045-3059 ◽  
Author(s):  
A Ghorbanpour Arani ◽  
M Mohammadimehr ◽  
A R Saidi ◽  
A Arefmanesh ◽  
Q Han
Keyword(s):  
Cylindrical Shell ◽  
Shell Theory ◽  
Buckling Load ◽  
Spring Constant ◽  
Single Walled Carbon Nanotube ◽  
Critical Buckling Load ◽  
Single Walled Carbon ◽  
Non Local ◽  
The Difference ◽  
Walled Carbon Nanotubes

In this article, the buckling analysis of a single-walled carbon nanotube using the non-local cylindrical shell theory under general loading embedded on the Winkler- and Pasternak-type foundations is presented. The effect of the surrounding elastic medium such as the Winkler-type spring constant and the Pasternak-type shear constant is taken into account in the present formulations. The non-local and local critical buckling loads are obtained under general loading such as the axial compression, lateral pressure, and torsional loading, and it is concluded from the results that the non-local critical buckling load under general loading is lower than the local critical buckling load. It is seen that the Winkler-type spring constant and Pasternak-type shear constant increase the non-local critical buckling load under general loading, therefore the difference between the presence and the absence of the Pasternak-type shear constant is large.

Buckling Analysis of Shells with a Circumferential Crack

2006 ◽  
Vol 306-308 ◽  
pp. 55-60
Author(s):  
I.S. Putra ◽  
T. Dirgantara ◽  
Firmansyah ◽  
M. Mora
Keyword(s):  
Boundary Conditions ◽  
Cylindrical Shells ◽  
Buckling Analysis ◽  
Numerical Results ◽  
Buckling Load ◽  
Numerical Analyses ◽  
Radius Of Curvature ◽  
Critical Buckling Load ◽  
Different Boundary Conditions ◽  
Circumferential Crack

In this paper, buckling analysis of cylindrical shells with a circumferential crack is presented. The analyses were performed both numerically using FEM and experimentally. The numerical analyses and experiments were conducted for several crack lengths and radius of curvature, and two different boundary conditions were applied, i.e. simply support and clamp in all sides. The results show the effect of the presence of crack to the critical buckling load of the shells. There are good agreements between experimental and numerical results.

Design of Ellipsoidal Heads Using Elastic-Plastic Finite Element Analysis

2002 ◽  
Author(s):  
Donald J. Florizone
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Pressure Vessel ◽  
Large Strain ◽  
Large Diameter ◽  
Spherical Shape ◽  
Element Analysis ◽  
Perfectly Plastic ◽  
Section Viii ◽  
Design Calculations

Traditional design techniques result in excess material being required for ellipsoidal heads. The 2001 ASME Boiler and Pressure Vessel Code Section VIII Division 1, UG-32D and Section VIII Division 2, AD-204 limit the minimum design thickness of the heads. ASME Boiler and Pressure Vessel Code Case 2261 provides alternate equations that enable thinner head design thickness. VIII-2 Appendix 3 and 4 methods potentially could be used to further optimize the head thickness. All the equations in the code use one thickness for the entire head. On large diameter thin heads the center or spherical area is often thicker than the knuckle area due to the method of manufacture. Including this extra material in the design calculations results in an increase of the MAWP of large diameter thin heads. VIII-2, AD-200 of the code permits localized thinning in a circumferential band in a cylindrical shell. Applying these same rules to elliptical heads would permit thinning in the knuckle region as well. Engineers have powerful finite element analysis tools that can be used to accurately determine levels of plastic strain and plastic deformed shapes. It is proposed that VIII-2 Appendix 4 and 5 methods be permitted for the design of elliptical heads. Doing so would permit significant decreases in thickness requirements. Different methods of Plastic Finite Element Analysis (PFEA) are investigated. An analysis of a PVRC sponsored burst test is done to develop and verify the PFEA methods. Two designs based on measurements of actual vessels are analyzed to determine the maximum allowable working pressures (MAWP) for thick and thin heads with and without local thin regions. MAWP is determined by limit analysis, per VIII-2 4-136.3 and by two other proposed methods. Using Burst FEA, the calculated burst pressure is multiplied by a safety factor to obtain MAWP. Large deflection large strain elastic perfectly plastic limit analyses (LDLS EPP LL) method includes the beneficial effect of deformations when determining the maximum limit pressure. Elliptical heads become more spherical during deformation. The spherical shape has higher pressure restraining capabilities. An alternate design equation for elliptical heads based on the LDLS EPP LL calculations is also proposed.

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